32. What Do You Want That Definition For, Anyway?

I have lost track of the number of times I've been in a discussion, and it's already circling a drain called "arguments about definition". It's a terrifyingly powerful vortex to get stuck in, whether you're sailing alone or with a crew; many a bold mathematician has been sucked into it, never to return. It's the spiral of trying to figure out whether a hot dog counts as a sandwich, or whether a tree falling in the forest makes a sound, or whether abortion is murder, or whether 1 should be a prime number.

My answer to all such questions these days is one and the same: "what do you want to know that for?" Any working mathematician or philosopher worth their salt will tell you that proofs and theorems are flashy and beautiful and nice and all, and that the choice of axioms backing them up is critical, but the most important thing of all is your choice of definitions. Those definitions, in turn, need to capture something important about some concept or phenomenon you want to reason about. If you care a lot about natural numbers, then you'd better be able to define 0; if you care about jurisprudence, then you'd better be able to operationalize "precedent".

Failing at this is a mark of unskilled conceptual engineering, on the other hand: a major indicator here is definitions which don't cut reality at all. As I sometimes put it, what a definition doesn't pick out can be more important than what it does; good and useful concepts are defined by their negative space. If your definition of "agent" is perfectly happy to call anything at all - a rock, a piece of paper, empty space - an agent, then your definition of "agent" is badly constructed and you should scrap it and try again.

But if it works well? If the definition is not just solid, but serves a useful function in conjunction with other definitions, and helps to assemble the foundations for a larger idea, or paint a clearer picture of some aspect of the world? Then that's conceptual engineering going right, and you can know it on sight. A gear is defined partially by its cogs, and partially by the empty space between the cogs, but most of all by the geartrain in which it plays a crucial and irreplaceable part: so too with definitions. There's a type theory sort of flavor to it, or a whiff of compositionality - a definition, an engineered concept, a mental object... ought to be shaped by what feeds into it and what it feeds into in turn.

The thing to internalize is that conceptual engineering requires a sort of boldness, a confidence in your own judgement to point at a definition and call it bad, along with a mirroring arrogance to feel like you've been licensed to slap together an operationalization and claim it reflects something important about the world. You are called on to make value judgements about abstract or mental objects that feel weird to handle and play around with, and that can be scary, but it's necessary. Seize the permission for yourself - no one can stop you. The crucial thing in making the decision is always - what do you want it for? What purpose does this definition need to serve? What other definitions, what other mental objects, does the definition need to mesh well with? Figure that out, and you figure out what you should make of your test cases.

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